Perform the row operation, $R_1-5R_2\rightarrow R_1$, on the following matrix. $\left[\begin{array} {ccc} 3 & 2 & 6 & 9 \\ 1 & -4 & 7 & 5 \\ 3 & 4 & 6 & 1 \end{array} \right] $
Answer: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_1$ and $R_2$ are given below. $R_1=\left[\begin{array} {ccc} 3 & 2 & 6 & 9 \end{array} \right] ~~~~~ R_2=\left[\begin{array} {ccc} 1 & -4 & 7 & 5 \end{array} \right]$ We are asked to perform the row operation, $R_1-5R_2\rightarrow R_1$. Therefore, we must add $-5R_2$ to $R_1$. $\begin{aligned}R_1-5R_2 &= \left[\begin{array} {ccc} 3 & 2 & 6 & 9 \end{array} \right] - 5\left[\begin{array} {ccc} 1 & -4 & 7 & 5\end{array} \right] \\\\&=\left[\begin{array} {ccc} -2 & 22 & -29 & -16 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_1$ with $R_1-5R_2$. $\left[\begin{array} {ccc} {3} & {2} & {6} & {9} \\ 1 & -4 & 7 & 5 \\ 3 & 4 & 6 & 1 \end{array} \right] \xrightarrow{R_1-5R_2\rightarrow R_1} \left[\begin{array} {ccc} {-2} & {22} & {-29} & {-16} \\ 1 & -4 & 7 & 5 \\ 3 & 4 & 6 & 1 \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -2 & 22 & -29 & -16 \\ 1 & -4 & 7 & 5 \\ 3 & 4 & 6 & 1 \end{array} \right]$